Thanks for sharing the details.
I checked all three website link you have shared and can confirm that post carousel is not displaying.
Upon further checking I see that you are using fairly old version of X Theme and Cornerstone other then the 3rd link in which you are using latest version of Cornerstone. Please note that all three websites version of Cornerstone and X Theme are incompatible. In that regards I would first suggest you to update X Theme and Cornerstone to version 6.2.2 and 3.2.2 respectively. Before updating the live website, I would suggest you to copy the same to a staging server and update over there. In case any issue comes up, same can be looked at without effecting the live website.
To update Cornerstone and X Theme please take a look at following resources:
Version compatibility guide: To learn about the latest and compatible versions of bundled and supported plugins, please take a look at following doc:
2. Theme Update guide: Please take a look at following article to update Cornerstone.
3. Please take a look at following doc about changelog:
_After doing the updates, always remember to clear all caches (if you are using WP Rocket, WP SuperCache or W3 Total Cache) when updating so that the code from the latest release is always in use. This will help you to avoid any potential errors.-